3,657 research outputs found

    On Hamilton Decompositions of Line Graphs of Non-Hamiltonian Graphs and Graphs without Separating Transitions

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    In contrast with Kotzig's result that the line graph of a 33-regular graph XX is Hamilton decomposable if and only if XX is Hamiltonian, we show that for each integer k≥4k\geq 4 there exists a simple non-Hamiltonian kk-regular graph whose line graph has a Hamilton decomposition. We also answer a question of Jackson by showing that for each integer k≥3k\geq 3 there exists a simple connected kk-regular graph with no separating transitions whose line graph has no Hamilton decomposition

    Long path and cycle decompositions of even hypercubes

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    We consider edge decompositions of the nn-dimensional hypercube QnQ_n into isomorphic copies of a given graph HH. While a number of results are known about decomposing QnQ_n into graphs from various classes, the simplest cases of paths and cycles of a given length are far from being understood. A conjecture of Erde asserts that if nn is even, â„“<2n\ell < 2^n and â„“\ell divides the number of edges of QnQ_n, then the path of length â„“\ell decomposes QnQ_n. Tapadia et al.\ proved that any path of length 2mn2^mn, where 2m<n2^m<n, satisfying these conditions decomposes QnQ_n. Here, we make progress toward resolving Erde's conjecture by showing that cycles of certain lengths up to 2n+1/n2^{n+1}/n decompose QnQ_n. As a consequence, we show that QnQ_n can be decomposed into copies of any path of length at most 2n/n2^{n}/n dividing the number of edges of QnQ_n, thereby settling Erde's conjecture up to a linear factor

    Resolution of the Oberwolfach problem

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    The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of K2n+1K_{2n+1} into edge-disjoint copies of a given 22-factor. We show that this can be achieved for all large nn. We actually prove a significantly more general result, which allows for decompositions into more general types of factors. In particular, this also resolves the Hamilton-Waterloo problem for large nn.Comment: 28 page

    Hamilton cycles in graphs and hypergraphs: an extremal perspective

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    As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page limits, this final version is slightly shorter than the previous arxiv versio

    A graph partition problem

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    Given a graph GG on nn vertices, for which mm is it possible to partition the edge set of the mm-fold complete graph mKnmK_n into copies of GG? We show that there is an integer m0m_0, which we call the \emph{partition modulus of GG}, such that the set M(G)M(G) of values of mm for which such a partition exists consists of all but finitely many multiples of m0m_0. Trivial divisibility conditions derived from GG give an integer m1m_1 which divides m0m_0; we call the quotient m0/m1m_0/m_1 the \emph{partition index of GG}. It seems that most graphs GG have partition index equal to 11, but we give two infinite families of graphs for which this is not true. We also compute M(G)M(G) for various graphs, and outline some connections between our problem and the existence of designs of various types

    Primitive decompositions of Johnson graphs

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    A transitive decomposition of a graph is a partition of the edge set together with a group of automorphisms which transitively permutes the parts. In this paper we determine all transitive decompositions of the Johnson graphs such that the group preserving the partition is arc-transitive and acts primitively on the parts.Comment: 35 page
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